MODULI SPACES OF CONNECTIONS ON A RIEMANN SURFACE
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1 MODULI SPACES OF CONNECTIONS ON A RIEMANN SURFACE INDRANIL BISWAS AND VICENTE MUÑOZ Abstract. Let E be a holomorphic vector bundle over a compact connected Riemann surface X. The vector bundle E admits a holomorphic projective connection if and only if for every holomorphic direct summand F of E of positive rank, the equality degree(e)/rank(e) = degree(f )/rank(f ) holds. Fix a point x 0 in X. There is a logarithmic connection on E, singular over x 0 with residue d n Id E x0 if and only if the equality degree(e)/rank(e) = degree(f )/rank(f ) holds. Fix an integer n 2, and also fix an integer d coprime to n. Let M(n, d) denote the moduli space of logarithmic SL(n, C) connections on X singular of x 0 with residue d n Id. The isomorphism class of the variety M(n, d) determines the isomorphism class of the Riemann surface X. 1. Introduction Let X be a compact connected Riemann surface. Let E be a holomorphic vector bundle over X of rank n. A holomorphic connection on E is given by locally defined holomorphic trivializations of E such that all the transition functions are locally constant maps to GL(n, C). We recall that if {U i } i I is a covering of X by open subsets and ϕ i : E Ui U i C n are holomorphic isomorphisms, then for an ordered pair (i, j) I I, the transition function g i,j : U i U j GL(n, C) is the unique function satisfying the identity g i,j ϕ i = ϕ j. Therefore, if E is equipped with a holomorphic connection, then it makes sense to talk of locally constant holomorphic sections of E. A holomorphic projective connection on E is given by locally defined holomorphic trivializations of E such that all the transition functions g i,j project to locally constant maps under the projection GL(n, C) PGL(n, C). Therefore, if E is equipped with a holomorphic projective connection, then the projective bundle P(E) has locally defined holomorphic trivializations such that all the transition functions are locally constant maps to PGL(n, C). Fix a point x 0 X. Set X := X \ {x 0 } to be the complement. A logarithmic connection on E singular at x 0 is a holomorphic connection on E X which, locally around x 0, has a holomorphic connection matrix with a single pole at x 0. Given a logarithmic 2000 Mathematics Subject Classification. 14H60, 14C30. 1
2 2 I. BISWAS AND V. MUÑOZ connection on E singular at x 0, the behavior of the flat sections near x 0 are captured by what is called the residue of the connection. The residue is a linear endomorphism of the fiber E x0. If the residue of a logarithmic connection on E singular at x 0 is a scalar multiple of the identity automorphism of E x0, then the logarithmic connection gives a holomorphic projective connection on E. Here we shall consider logarithmic connections whose residue is a scalar multiple of the identity automorphism. We investigate some properties of holomorphic projective connections and logarithmic connections. Especially we address the question of existence of such connections on a given holomorphic vector bundle. In the last section we explain a recent result of the authors on the moduli spaces of logarithmic connections. 2. Connections and projective connections Let X be a compact connected Riemann surface; alternatively, X is an irreducible smooth projective curve defined over the field of complex numbers. The complexified tangent bundle T R X R C decomposes as T R X R C = T 1,0 X T 0,1 X into (1, 0) and (0, 1) components. The dual line bundle (T 1,0 X) will also be denoted by Ω 1,0 X or K X, and the complex line bundle (T 0,1 X) will also be denoted by Ω 0,1 X. A holomorphic vector bundle over X is a C vector bundle of rank n whose transition functions are given by holomorphic maps to GL(n, C). The usual Dolbeault operator -operator on locally defined smooth functions over X give a Dolbeault operator E on a holomorphic vector bundle E. This operator E is a first order differential operator that sends smooth sections of E to those of E Ω 0,1 X. We note that the line bundles T 1,0 X and K X have natural holomorphic structures. The trivial holomorphic line bundle X C will also be denoted by O X. Definition 2.1. Let E be a holomorphic vector bundle over X of rank n. A holomorphic connection on E is a first order holomorphic differential operator D : E E K X satisfying the Leibniz identity, which says that D(fs) = fd(s) + df s, where f (respectively, s) is a locally defined holomorphic function (respectively, holomorphic section of E). Associated to a holomorphic connection D there is a C connection : C (E) C (E T X),
3 MODULI SPACE OF CONNECTIONS 3 where C (E) denotes the sheaf of smooth sections of the vector bundle E. Using the isomorphism Γ(E) = O(E) O C, where O(E) is the sheaf of holomorphic sections of E, we have (fs) = fd(s) + df s, for s a holomorphic section of E and f C (X). In other terms, we define = D + E on holomorphic sections, and then we extend to C sections. Let D be a holomorphic connection on E. The curvature D D of D is a holomorphic section of the vector bundle End(E) Ω 2,0 X. Since dim C X = 1, we have Ω 2,0 X = 0. Hence a holomorphic connection on a Riemann surface is automatically flat. If E has a holomorphic connection, then degree(e) = 0 [At]. Actually, the connection is also flat, since its curvature is F = D E + E D, which is zero on holomorphic sections, hence F = 0. Considering a local flat trivialization, the constant sections (in the local trivialization) are automatically holomorphic, hence there are local flat holomorphic trivializations of E. Note also that any flat connection on a C vector bundle E gives rise to a holomorphic structure on E together with a holomorphic connection. For this, we only need to take flat trivializations and declare them to be holomorphic. Let E be a holomorphic vector bundle over X and F a direct summand of E. This means that F is a holomorphic subbundle of E, and there exists a holomorphic subbundle F of E such that the natural homomorphism F F E is an isomorphism. Let ι : F E be the inclusion map. Fix a complement F of F as above. Using the natural isomorphism F F E we get a projection q : E F. If D is a holomorphic connection on E, then it is easy to see that the composition F ι E D E K X q Id F K X is a holomorphic connection on F. Hence if E admits a holomorphic connection, then each direct summand of E also admits a holomorphic connection. So, if E admits a holomorphic connection, then the degree of each direct summand of E is zero. A theorem due to Atiyah and Weil says that the converse is also true. Theorem 2.2 ([At], [We]). A holomorphic vector bundle E over X admits a holomorphic connection if and only if the degree of each direct summand of E is zero. In particular, any indecomposable bundle E admits a holomorphic connection. A holomorphic vector bundle E of rank n over X admits a holomorphic connection if and only if we can choose (holomorphic) local trivializations of E over a covering of
4 4 I. BISWAS AND V. MUÑOZ X (by open sets) such that all the transition functions are locally constant functions to GL(n, C). This follows from the flatness of. Recall that the Atiyah bundle At(E) associated to a holomorphic vector bundle E is the holomorphic vector bundle associated to the sheaf of first order holomorphic differential operators on E. The Atiyah exact sequence for E is (2.1) 0 End(E) At(E) T 1,0 X 0 where the last arrow is the symbol map. Note that End(E) is identified with the zeroth order differential operators. Therefore, a holomorphic connection on E corresponds to a holomorphic splitting of the Atiyah exact sequence in (2.1). Now let P E be the holomorphic principal GL(n, C) bundle over X defined by E. So P E is the space of all bases in the fibers of E. Let O X End(E) be the line subbundle given by the sheaf of endomorphisms of E of the type s f s. Therefore, from (2.1) we have the exact sequence of vector bundles (2.2) 0 End(E)/O X At(E)/O X T 1,0 X 0 over X. This exact sequence is the Atiyah bundle for the principal PGL(n, C) bundle over X corresponding to E. We note that this principal PGL(n, C) is the quotient space P E /C, where C is considered as a subgroup of GL(n, C) through scalar multiplications of C n. Also, note that the quotient vector bundle End(E)/O X is identified with the subbundle of ad(e) End(E) defined by trace zero endomorphisms. Indeed, the natural projection of the subbundle ad(e) to the quotient bundle End(E)/O X is an isomorphism. Definition 2.3. A holomorphic projective connection on E is a holomorphic splitting of the exact sequence in (2.2). By a projective connection we will always mean a holomorphic projective connection. It follows that a holomorphic vector bundle E admits a projective connection if we can choose holomorphic local trivializations of E over a covering of X (by open sets) such that images of all the transition functions are locally constant under the projection GL(n, C) PGL(n, C). This means that they are of the form f Id n n times a locally constant function, with f being some nonzero holomorphic function. To see this, consider the splitting of the exact sequence (2.2) given by the projective connection, and take a covering by open sets U α. Over each U α, take an arbitrary splitting of At(E) At(E)/O X. This produces a holomorphic connection D α on U α, and hence a flat trivialization of E Uα. Since D α D β = f αβ Id n n on U α Uβ, we have that the transition functions of E (with respect to the chosen trivializations) are of the form f Id n n times a locally constant function. Assume that the vector bundle E admits a projective connection. Let E PGL be the principal PGL(n, C) bundle given by E. We noted earlier that E PGL = P E /C, where P E is the principal GL(n, C) bundle given by E. Giving a projective connection on E is
5 MODULI SPACE OF CONNECTIONS 5 equivalent to giving a holomorphic connection on the principal PGL(n, C) bundle E PGL, that is, a family of flat trivializations of E PGL such that the transition functions are locally constant functions with values in PGL(n, C). Note that this also can be expressed in terms of giving trivializations of the projective bundle P(E) whose transition functions are locally constant functions in PGL(n, C). Using the commutative diagram of vector bundles 0 End(E) At(E) T 1,0 X 0 0 End(E)/O X At(E)/O X T 1,0 X 0 over X, a holomorphic connection on E induces a projective connection on E. However, the converse is not true in the sense that a vector bundle admitting a projective connection need not admit a holomorphic connection. If E admits a projective connection and degree(e) = 0 then the projective connection is induced by a holomorphic connection. To see this, take a covering by open sets U α with a holomorphic connection D α on each U α, as before. Since D α D β = f αβ Id n n on U α Uβ, we have that f αβ Ω 1 X (U α Uβ ) form a cocycle, and hence define a class in H 1 (X, K X ) = C. Therefore there exists g α Ω 1 X (U α) such that if we modify D α to D α = D α +g α Id n n, then the corresponding D α D β is a constant multiple of the identity. This gives transition functions for E which are locally constant functions in GL(n, C). The following theorem classifies all holomorphic vector bundles over X that admit a projective connection. Theorem 2.4. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following two statements are equivalent: (1) The vector bundle E admits a projective connection. (2) For any direct summand F E, the equality holds. degree(f ) rank(f ) Proof. Assume that the vector bundle E admits a projective connection. Let E PGL be the principal PGL(n, C) bundle given by E. The projective connection on E gives a holomorphic connection on E PGL. The adjoint vector bundle over X for the principal PGL(n, C) bundle E PGL is ad(e). Therefore ad(e) admits a family of trivializations with transitions functions locally constant functions on PGL(n, C). This means that ad(e) admits a projective connection. Since the degree of ad(e) is zero, the vector bundle ad(e) admits a holomorphic connection. = d n Let D be a holomorphic connection on ad(e). Assume that E = F 1 F 2.
6 6 I. BISWAS AND V. MUÑOZ We will show that D induces a holomorphic connection on the vector bundle Hom(F 1, F 2 ). For this, first note that Hom(F 1, F 2 ) is a direct summand of ad(e). Indeed, we have a holomorphic decomposition (2.3) ad(e) = Hom(F 1, F 2 ) Hom(F 2, F 1 ) (End(F 1 ) End(F 2 ))/O X of ad(e) into a direct sum of vector bundles; here O X is considered as a subbundle of End(F 1 ) End(F 2 ) by sending any locally defined holomorphic function f to f(id F1 + Id F2 ). Hom(F 1, F 2 ) is a direct summand of the bundle ad(e), which admits a holomorphic connection. Hence Theorem 2.2 implies that (2.4) degree(hom(f 1, F 2 )) = n 1 d 2 d 1 n 2 = 0, where n i (respectively, d i ) is the rank (respectively, degree) of F i, i = 1, 2. Since from (2.4), it follows immediately that degree(e) = d 1 + d 2 = d, degree(f i ) rank(f i ) = d n. Therefore, the first statement in the theorem implies the second statement. To prove the converse, we begin with a lemma. Lemma 2.5. Let E 1 and E 2 are two holomorphic vector bundles over X, both admitting projective connections. If degree(e 1 ) = degree(e 2) rank(e 1 ) rank(e 2 ), then E 1 E2 also admits a projective connection. Proof. Since both E 1 and E 2 admit projective connections, the vector bundle E 1 E2 also admits a projective connection. On the other hand, the condition that degree(e 1 ) rank(e 1 ) = degree(e 2) rank(e 2 ) implies that degree(e 1 E2 ) = 0. This condition together with the condition that E 1 E2 admits a projective connection imply that E 1 E2 admits a holomorphic connection. Similarly, E 2 E1 admits a holomorphic connection. Now using the natural decomposition End(E 1 E 2 ) = End(E 1 ) End(E 2 ) (E 1 E 2 ) (E 2 E 1 ) it follows that E 1 E2 also admits a projective connection, and the proof of the lemma is complete. Continuing with the proof of the theorem, assume that for each direct summand F of E. degree(f ) rank(f ) = d n
7 MODULI SPACE OF CONNECTIONS 7 We will prove that E admits a projective connection. This will be done by imitating the Atiyah s proof in [At] of the criterion for the existence of a holomorphic connection. As before, let E PGL denote the holomorphic principal PGL(n, C) bundle over X defined by E. Let (2.5) 0 ad(e) At(E PGL ) T 1,0 X 0 be the Atiyah exact sequence for the principal PGL(n, C) bundle E PGL (see [At, page 187, Theorem 1]), where At(E PGL ) is the Atiyah bundle for E PGL. The exact sequence of vector bundles in (2.5) coincides with the exact sequence in (2.2). A projective connection on E is equivalent to a holomorphic connection on the principal PGL(n, C) bundle E PGL. Let (2.6) a(e) H 1 (X, K X ad(e)) be the obstruction class for holomorphic splitting of (2.5). We note that the vector bundle ad(e) is self-dual, that is, ad(e) = ad(e). Indeed, the trace map ad(e) ad(e) O X that sends s t to trace(s t) identifies ad(e) with ad(e). Therefore, the Serre duality says that Let H 1 (X, K X ad(e)) = H 0 (X, ad(e)). (2.7) b(e) H 0 (X, ad(e)) be the linear form corresponding to the cohomology class a(e), constructed in (2.6), by the Serre duality. In view of Lemma 2.5, it suffices to prove the theorem under the assumption that the vector bundle E is indecomposable. Assume that E is indecomposable. Then any section s H 0 (X, ad(e)) is actually nilpotent [At, page 201, Proposition 16]. In other words, the endomorphism s gives a holomorphic filtration of subbundles 0 = E 0 E 1 E 2 E k = E such that s(e i ) = E i 1 for all 1 i k. More precisely, E i is the subbundle of E generated by the kernel of the i-fold composition s i = s s. Now, for a nilpotent endomorphism s of E we know that b(e)(s) = 0, where b(e) is constructed in (2.7) [At, page 202, Proposition 18(ii)]. Since the functional b(e) vanishes on H 0 (X, ad(e)), we have b(e) = 0. Therefore, the obstruction class a(e)
8 8 I. BISWAS AND V. MUÑOZ in (2.6) vanishes, and hence E PGL admits a holomorphic connection. This completes the proof of the theorem. 3. Logarithmic connections As before, X will denote a compact connected Riemann surface of genus g, where g 2. Fix a point x 0 X. The holomorphic line bundle over X defined by the divisor x 0 will be denoted by O X (x 0 ). Definition 3.1. Let E be a holomorphic vector bundle over X. A logarithmic connection on E singular over x 0 is a holomorphic differential operator satisfying the Leibniz identity D : E E O X (x 0 ) K X (3.1) D(fs) = fd(s) + df s, where f (respectively, s) is a locally defined holomorphic function (respectively, holomorphic section of E). Note that a logarithmic connection on E singular over x 0 produces a holomorphic connection on E over X \ {x 0 }. Let U be a chart around x 0 with local holomorphic coordinate z. Then (3.2) D(s) = ds + A(z)s dz z, for any local holomorphic section s of End(E) and a holomorphic connection matrix A(z). The curvature of a holomorphic connection on E is a holomorphic two form with values in End(E) O X (x 0 ). Since a Riemann surface does not have nonzero holomorphic two forms, any logarithmic connection on a Riemann surface is flat. A differential operator that satisfies the Leibniz identity (3.1) is clearly of order one. The above condition that a logarithmic connection D satisfies the Leibniz identity is evidently equivalent to the condition that the symbol of the first order differential operator D coincides with the section Id E H 0 (X, O X (x 0 ) End(E)), where Id E denotes the identity automorphism of E. The Poincaré adjunction formula says the following: Let D be a smooth hypersurface on a smooth variety Z, and let L denote the line bundle over Z defined by the divisor D. Then the restriction of the line bundle L to D is canonically identified with the normal bundle of D. See [GH] for a proof. Therefore, the Poincaré adjunction formula says that the fiber over x 0 of the line bundle O X (x 0 ) is identified with the holomorphic tangent space T x0 X. Using this isomorphism between O X (x 0 ) x0 and T x0 X, the fiber (K X OX (x 0 )) x0 is identified with C.
9 MODULI SPACE OF CONNECTIONS 9 We will now recall the definition of the important notion of residue of a logarithmic connection. Let D be a logarithmic connection on a vector bundle E over X, which is singular over x 0. Let v E x0 be any vector in the fiber of E over the point x 0. Let v be a holomorphic section of E defined around x 0 such that v(x 0 ) = v. Consider D( v)(x 0 ) (K X O X (x 0 )) x0 C E x0 = C C E x0 = E x0. Note that if v = 0, then D( v) is a (locally defined) section of the subsheaf E K X E O X (x 0 ) K X. So, in that case the above evaluation D( v)(x 0 ) E x0 vanishes. Consequently, we have a well defined endomorphism Res(D, x 0 ) End(E x0 ) that sends any v E x0 to D( v)(x 0 ) E x0. This endomorphism Res(D, x 0 ) is called the residue of the logarithmic connection D at the point x 0. See [De1] for more details. If D is a logarithmic connection on E singular over x 0 and θ H 0 (X, End(E) K X ), then the differential operator D + θ is also a logarithmic connection on E singular over x 0. Furthermore, we have Res(D, x 0 ) = Res(D + θ, x 0 ). Conversely, if D and D are two logarithmic connections on E regular on X \ {x 0 } with (3.3) Res(D, x 0 ) = Res(D, x 0 ), then D = D + θ, where θ H 0 (X, End(E) K X ). In this way the space of all logarithmic connections D on the given vector bundle E, regular on X\{x 0 } and satisfying (3.3), with D fixed, is an affine space for the vector space H 0 (X, K X End(E)). Let E be a holomorphic vector bundle over X of rank n, and let D be a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition (3.4) Res(D, x 0 ) = d n Id E x0, where d C. Consider the nonsingular flat connection on the complement X \ {x 0 } defined by D. The above condition on the residue implies that the monodromy around x 0 of this flat connection is the n n diagonal matrix with exp(2π 1d/n) as the diagonal entries [De1, page 79, Proposition 3.11]. From the expression of the degree of E in terms of the residue of D it follows immediately that degree(e) = d; the last sentence of Corollary B.3 in [EV, page 186] gives the expression of the first Chern class in terms of the trace of the residue. In particular, d in (3.4) must be an integer. We will now relate logarithmic connections with projective connections discussed in Section 2. Let E be a holomorphic vector bundle over X of rank n and degree d. Let D be a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition in
10 10 I. BISWAS AND V. MUÑOZ (3.4). The logarithmic connection D gives a holomorphic connection on the vector bundle E X\{x0 } over the open subset X \ {x 0 } X. This connection on E X\{x0 } induces a projective connection on the PGL(n, C) principal bundle E PGL X\{x0 } over X \{x 0 } X, understood as a family of (holomorphic) trivializations with transition functions which are locally constant in PGL(n, C). The residue of D is in the center of M(n, C) (the Lie algebra of GL(n, C)); its projection to sl(n, C) vanishes. Therefore, the projective connection of E PGL X\{x0 } extends across x 0 as a regular projective connection. Hence we have proved the following lemma: Lemma 3.2. Let E be a holomorphic vector bundle over X such that E admits a logarithmic connection D regular on X \ {x 0 } such that Res(D, x 0 ) = d n Id E x0. Then the vector bundle E admits a projective connection. The following theorem is the analog of Theorem 2.4 for logarithmic connections. Theorem 3.3. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following two statements are equivalent: (1) The vector bundle E admits a logarithmic connection regular on X \ {x 0 } whose residue at x 0 is d n Id E x0. (2) For any direct summand F E, the equality holds. degree(f ) rank(f ) Proof. Let D be a logarithmic connection on the vector bundle E, regular on X\{x 0 }, with residue Res(D, x 0 ) = d n Id E x0. From Lemma 3.2 we know that E admits a projective connection. Therefore, from Theorem 2.4 it follows that for every direct summand F of E. To prove the converse, assume that (3.5) degree(f ) rank(f ) degree(f ) rank(f ) for every direct summand F of E. Therefore, E admits a projective connection (see Theorem 2.4). Fix a projective connection D P on E. Let = d n = d n = d n (3.6) γ : X X
11 MODULI SPACE OF CONNECTIONS 11 be a ramified Galois covering of degree n with the property that that the map γ is totally ramified over the point x 0 X. (The map γ is allowed to have ramifications over points in X \{x 0 }.) The condition that γ is totally ramified over x 0 means that there is a unique point y 0 X such that γ(y 0 ) = x 0. Consider the vector bundle (3.7) V := O X( dy 0 ) O X γ E over X, where γ is the map in (3.6), and y 0 is the unique point with γ(y 0 ) = x 0. The projective bundle P(V ) over X is canonically identified with the projective bundle γ P(E). Therefore, the projective connection D P on E gives a projective connection γ D P on V. Note that degree(v ) = 0. Therefore, the condition that V admits a projective connection implies that V admits a holomorphic connection. Fix a holomorphic connection D on V. Let Γ denote the Galois group for the Galois covering γ in (3.6). Note that the Galois action of Γ on X has a canonical lift to the vector bundle V. Consider D = 1 h D #Γ which is a holomorphic connection on V (as the space of holomorphic connections on V is an affine space, the average of D over the action of the elements of Γ is again a holomorphic connection on V ). It is now straight forward to check that the Galois invariant connection D on V descends to a logarithmic connection on E. This descended logarithmic connection is regular outside x 0, and its residue at x 0 is d Id n E x0. This completes the proof of the theorem. Theorem 2.4 and Theorem 3.3 together have the following corollary: Corollary 3.4. Let E be a holomorphic vector bundle over X of rank n and degree d. Then the following three statements are equivalent: h Γ (1) The vector bundle E admits a logarithmic connection regular on X \ {x 0 } whose residue at x 0 is d n Id E x0. (2) The vector bundle E admits a projective connection. (3) For any direct summand F E, the equality degree(f ) rank(f ) = d n holds.
12 12 I. BISWAS AND V. MUÑOZ 4. Moduli space of logarithmic connections Henceforth, we will assume the rank n and the degree d are mutually coprime. Let M D (n) denote the moduli space of all logarithmic connections (E, D) over X, where E is any holomorphic vector bundle of rank n and degree d and D is a logarithmic connection on E, regular on X \ {x 0 }, satisfying the residue condition Res(D, x 0 ) = d n Id E x0. (See [Si1], [Si2], [Ni] for the construction of this moduli space.) So M D (n) parametrizes isomorphism classes of pairs of the form (E, D), where E is any rank n holomorphic vector bundle of degree d and D is a logarithmic connection on E regular on X \ {x 0 } with Res(D, x 0 ) = d Id n E x0. We note that every logarithmic connection over X occurring in the moduli space M D (n) is irreducible. Indeed, for (E, D) M D (n), if F E is a holomorphic subbundle invariant by the connection D, i.e. D(F ) F O(x 0 ) K X, then D induces a logarithmic connection D F on F which is regular on X \ {x 0 } and Res(D F, x 0 ) = d n Id F x0. Therefore, degree(f ) = d rank(f ) n. This contradicts the assumption that d and n are mutually coprime if F is a proper subbundle of E. Since every logarithmic connection over X occurring in the moduli space M D (n) is irreducible, the variety M D (n) is smooth; singular points of a moduli space of connections correspond to reducible connections. The variety M D (n) is known to be irreducible. The dimension of M D (n) is 2(n 2 (g 1) + 1). Consider the holomorphic line bundle L := O X (d x 0 ) over X. The de Rham differential f df, defines a logarithmic connection D L on L which is regular on X \ {x 0 }, and (4.1) Res(D L, x 0 ) = d Id Lx0. Let (4.2) M D (L) M D (n) be the subvariety parametrizing isomorphism classes of logarithmic connections (E, D) M D (n) such that n E = L, and the logarithmic connection on L induced by D using an isomorphism n E L coincides with the logarithmic connection D L in (4.1). Note that a logarithmic connection D on E induces a logarithmic connection on n E. Since any two holomorphic isomorphisms between n E and L differ by a constant scalar,
13 MODULI SPACE OF CONNECTIONS 13 the connection of L given by the induced connection on n E using an isomorphism n E L is independent of the choice of the isomorphism. The subset M D (L) in (4.2) is an irreducible smooth closed subvariety of dimension 2(n 2 1)(g 1). We will show that the biholomorphism class of both M D (L) and M D (n) are independent of the complex structure of X. Let X := X \ {x 0 } be the complement. Fix a point x X. The point x 0 gives a conjugacy class in the fundamental group π 1 (X, x ) as follows. Let f : D X be an orientation preserving embedding of the closed unit disk D := {z C z 2 1} into the Riemann surface X such that f(0) = x 0. The free homotopy class of the map D = S 1 X obtained by restricting f to the boundary of D is independent of the choice of f. The orientation of D coincides with the anti clockwise rotation around x 0. Any free homotopy class of oriented loops in X gives a conjugacy class in π 1 (X, x ). Let γ denote the orbit in π 1 (X, x ), for the conjugation action of π 1 (X, x ) on itself, defined by the above free homotopy class of oriented loops associated to x 0. Let (4.3) Hom 0 (π 1 (X, x ), GL(n, C)) Hom(π 1 (X, x ), GL(n, C)) be the space of all homomorphisms from the fundamental group π 1 (X, x ) to GL(n, C) satisfying the condition that the image of γ (the free homotopy class defined above) is exp(2π 1d/n) I n n. It may be noted that since exp(2π 1d/n) I n n is in the center of SL(n, C), a homomorphism sends the orbit γ in π 1 (X, x ) (for the adjoint action of π 1 (X, x ) on itself) to exp(2π 1d/n) I n n if and only if there is an element in the orbit which is mapped to exp(2π 1d/n) I n n. Take any homomorphism ρ Hom 0 (π 1 (X, x ), GL(n, C)). Let (V, ) be the flat vector bundle or rank n over X given by ρ. Therefore V is a holomorphic vector bundle on X. The monodromy of along the oriented loop γ is exp(2π 1d/n) I n n. Using the logarithm 2π 1d/n I n n of the monodromy, the vector bundle V over X extends to a holomorphic vector bundle V over X, and furthermore, the connection on V extends to a logarithmic connection on the vector bundle V over X such that (V, ) M D (n), where M D (n) is the moduli space of logarithmic connections defined earlier (see [Ma, p. 159, Theorem 4.4]). Since GL(n, C) is an algebraic group defined over the field of complex numbers, and π 1 (X, x ) is a finitely presented group, the representation space Hom(π 1 (X, x ), SL(n, C)) is a complex algebraic variety in a natural way. The conjugation action of GL(n, C) on itself induces an action of GL(n, C) on Hom 0 (π 1 (X, x ), GL(n, C)). The action of any T GL(n, C) on Hom 0 (π 1 (X, x ), GL(n, C)) sends any homomorphism ρ to the
14 14 I. BISWAS AND V. MUÑOZ homomorphism π 1 (X, x ) GL(n, C) defined by β T 1 ρ(β)t. Let (4.4) R g := Hom 0 (π 1 (X, x ), GL(n, C))/GL(n, C) be the quotient space for this action. The algebraic structure of Hom 0 (π 1 (X, x ), GL(n, C)) induces an algebraic structure on the quotient R g. The scheme R g is an irreducible smooth quasiprojective variety of dimension 2(n 2 1)(g 1) + 2 defined over C. The isomorphism class of this variety R g is independent of the complex structure of the topological surface X. The isomorphism class depends only on the integers g, n and d. On the other hand, the moduli space M D (n) is canonically biholomorphic to R g. Therefore, the biholomorphism class of the moduli space M D (n) is independent of the complex structure of X; the biholomorphism class depends only on the integers g, n and d. Replacing GL(n, C) by the algebraic subgroup SL(n, C) above, we have an algebraic irreducible smooth quasiprojective variety S g := Hom 0 (π 1 (X, x ), SL(n, C))/SL(n, C), which is biholomorphic to M D (L). We conclude that the biholomorphism class of the moduli space M D (L) defined in (4.2) is also independent of the complex structure of X. 5. The second intermediate Jacobian of the moduli space In this section we recall some results of [BM]. A holomorphic vector bundle E over X is called stable if for every nonzero proper subbundle F E, the inequality holds. degree(f ) rank(f ) < degree(e) rank(e) Let N X denote the moduli space parametrizing all stable vector bundles E over X with rank(e) = n and n E = L = O X (d x 0 ). The moduli space N X is an irreducible smooth projective variety of dimension (n 2 1)(g 1) defined over C. Let M D (L) be the moduli space defined in (4.2). Let (5.1) U M D (L) be the Zariski open subset parametrizing all (E, D) such that the underlying vector bundle E is stable. The openness of this subset follows from [Ma]. Let (5.2) Φ : U N X denote the forgetful map that sends any (E, D) to E. By Theorem 3.3, any E N X admits a logarithmic connection D such that (E, D) M D (L), since any E N X is indecomposable. Therefore, the projection Φ in (5.2) is
15 MODULI SPACE OF CONNECTIONS 15 surjective. Furthermore, Φ makes U an affine bundle over N X. More precisely, U is a torsor over N X for the holomorphic cotangent bundle T N X. This means that the fibers of the vector bundle T N X act freely transitively on the fibers of Φ [BR, p. 786]. Since M D (L) is irreducible, and U is nonempty, the subset U M D (L) is Zariski dense. The following lemma is proved in [BM]: Lemma 5.1. Let Z := M D (L) \ U be the complement of the Zariski open dense subset. The codimension of the Zariski closed subset Z in M X is at least (n 1)(g 2) + 1. For any i 0, the i th cohomology of a complex variety with coefficients in Z is equipped with a mixed Hodge structure [De2], [De3]. The following proposition is proved in [BM]. Proposition 5.2. The mixed Hodge structure H 3 (M D (L), Z) is pure of weight three. More precisely, the mixed Hodge structure H 3 (M D (L), Z) is isomorphic to the Hodge structure H 3 (N X, Z), where N X is the moduli space of stable vector bundles introduced at the beginning of this section. Let (5.3) J 2 (M D (L)) := H 3 (M D (L), C)/(F 2 H 3 (M D (L), C) + H 3 (M D (L), Z)) be the intermediate Jacobian of the mixed Hodge structure H 3 (M D (L)) (see [Ca, p. 110]). The intermediate Jacobian of any mixed Hodge structure is a generalized torus [Ca, p. 111]. Let J 2 (N X ) := H 3 (N X, C)/(F 2 H 3 (N X, C) + H 3 (N X, Z)) be the intermediate Jacobian for H 3 (N X, Z), which is a complex torus. The following proposition is proved in [BM]. Proposition 5.3. The intermediate Jacobian J 2 (M D (L)) is isomorphic to J 2 (N X ), which is isomorphic to the Jacobian Pic 0 (X) of the Riemann surface X. The homology H 1 (J 2 (M D (L)), Z) has a natural skew symmetric pairing which we will describe below. We first note that from Proposition 5.3, and the fact that H 3 (N X, Z) is torsionfree, it follows that (5.4) H 1 (J 2 (M D (L)), Z) = H 3 (M D (L), Z) = H 3 (N X, Z). Also, we have Fix a generator For any H 2 (M D (L), Z) = H 2 (N X, Z) = Z. α H 2 (M D (L), Z). (5.5) θ H 2(n 2 1)(g 1)(M D (L), Z),
16 16 I. BISWAS AND V. MUÑOZ we have a skew symmetric pairing (5.6) B θ : 2 H 3 (M D (L), Z) Z defined by B θ (ω 1, ω 2 ) = (ω 1 ω 2 α (n2 1)(g 1) 3 ) θ Z. The following theorem is proved in [BM]. Theorem 5.4. There exists a homology class θ as in (5.5) such that the pairing B θ in (5.6) is nonzero. Take any θ such that B θ is nonzero. Using the isomorphism in (5.4), B θ defines a nonzero pairing B θ : 2 H1 (J 2 (M D (L)), Z) Z. The pair (J 2 (M D (L)), B θ ) is isomorphic to Pic 0 (X) equipped with a multiple of the canonical principal polarization on Pic 0 (X) given by the class of a theta line bundle. Given a multiple of a principal polarization on an abelian variety, there is a unique way to recover the principal polarization from it. Therefore, from Theorem 5.4 it follows immediately that the isomorphism class of (Pic 0 (X), Θ), where Θ is the canonical polarization on Pic 0 (X), is determined by the isomorphism class of the variety M D (L). The Torelli theorem says that the isomorphism class of the principally polarized abelian variety (Pic 0 (X), Θ) determines the curve X up to an isomorphism. Thus Theorem 5.4 give the following corollary: Corollary 5.5. The isomorphism class of the variety M D (L) uniquely determines the isomorphism class of the curve X. This is in contrast with the fact that the biholomorphism class of the complex manifold M D (L) is independent of the complex structure of the Riemann surface X (see the end of Section 4). References [At] M. F. Atiyah, Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), [BM] I. Biswas and V. Muñoz, Torelli theorem for the moduli spaces of connections on a Riemann surface. Preprint. [BR] I. Biswas and N. Raghavendra, Line bundles over a moduli space of logarithmic connections on a Riemann surface. Geom. Funct. Anal. 15 (2005), [Ca] J. Carlson, Extensions of mixed Hodge structures, in: Journées de géométrie algébrique d Angers Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn Germantown, Md., 1980, pp [De1] P. Deligne, Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163, Springer Verlag, Berlin, [De2] P. Deligne, Théorie de Hodge, II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), [De3] P. Deligne, Théorie de Hodge, III. Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5 77.
17 MODULI SPACE OF CONNECTIONS 17 [EV] H. Esnault and E. Viehweg, Logarithmic de Rham complex and vanishing theorems. Invent. Math. 86 (1986), [GH] P. A. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, [Ma] M. Maruyama, Openness of a family of torsion free sheaves. Jour. Math. Kyoto Univ. 16 (1976), [Ni] N. Nitsure, Moduli of semistable logarithmic connections. Jour. Amer. Math. Soc. 6 (1993), [Si1] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79 (1994), [Si2] C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80 (1994), [We] A. Weil, Généralisation des fonctions abéliennes. Jour. Math. Pures Appl. 17 (1938), School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay , India address: indranil@math.tifr.res.in Departamento de Matemáticas, Consejo Superior de Investigaciones Científicas, Serrano 113 bis, Madrid, Spain address: vicente.munoz@imaff.cfmac.csic.es
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